Let's say we have two functions, $f(x)$ and $g(x)$, such that the Mellin transform of $f(x)$ converges on the strip $a < x < b$ and the Mellin transform of $g(x)$ coverges on the strip $c < x < \infty$ with $b<c$.
If we denote the analytic continuation of the Mellin transform of $g(x)$ as $G(x)$, is it true that $$\int_{0}^{\infty}(f(x)+g(x))x^{t-1}dx = \int_{0}^{\infty}f(x)x^{t-1}dx+G(t)$$ where the Mellin transform converges?